I recently visited Jeju from Anseong, and a few small planning choices made the trip much easier.

First, I would avoid trying to circle the whole island in one day. Jeju looks compact on a map, but driving between attractions takes longer than expected. It worked better to choose one area each day and keep two or three main stops.

Second, check the weather again in the morning. Conditions near Hallasan can be very different from the coast, so it helps to keep one indoor option ready. I also found that starting early made parking and meals much less stressful.

For anyone planning a first trip, I have been collecting practical Korea travel notes at https://www.koreafromalocal.com/. It is my own site, and I hope the local perspective is useful. If you have a Jeju question, feel free to ask here and I will answer if I can.